[[310,10,22]] weight-6 cyclic generalized-bicycle code over Z_155 (single circulant pair)

Direction & hypothesis

Cell: unrestricted x weight-6, base tier (n <= 700, no distance cap). The weight-6 cell holds few codes: its headline is [[672,20,32]] at kd^2/n 30.5 and below it sit [[630,12,34]], [[630,14,30]], [[576,12,30]], [[510,16,24]], [[540,12,28]], and [[360,12,24]], all at kd^2/n 18 to 22, then a long tail under 16. The hypothesis was that weight-3 polynomials in the bicycle families (cyclic GB, BB, 2BGA) still reach distances in the twenties and thirties at n = 200 to 700 once k is forced above 8, so that a strict Pareto record with kd^2/n above 12 is available in the band between those incumbents; the cell headline (30.5) was the stretch target.

What was searched

• Cyclic GB over odd m in [101, 349] (n = 2m in [202, 698]): a divisor g of x^m - 1 of degree 5 to 24 is chosen at random from the irreducible factors (sympy over GF(2)), every weight-3 multiple of g mod x^m - 1 containing x^0 is enumerated (residues x^i mod g, one lookup per pair), and pairs (a, b) from distinct rotation classes with gcd(offsets, m) = 1 are built with research/cyclic_gb.py build_cyclic_gb. k = 2 deg gcd(a, b, x^m - 1) >= 2 deg g; pairs with k > 2 deg g + 6 were dropped.
• 12,288 draws in the stream that produced this code; 11,951 passed the prefilter (CSS, exact k with gf2_fast, an eff floor of 12 on kd^2/n, and a strict-record threshold against the board checkout), 321 survived the first GPU stage, 263 the third. 2,610,200,000 GPU deep-kernel trials in total, 2.40 GPU hours busy on one NVIDIA A40.
• Screen: verify/ris_gpu.cu's deep kernel (full-basis RREF plus pair sums, pair depth 8) in three stages per side, 100,000, 500,000, and 2,000,000 trials, with early stop as soon as a logical lighter than the record threshold appears (sound, since RIS weights are upper bounds). Every recovered operator is re-verified with verify/gf2.py before it counts.
• Record thresholds were computed against the board checkout at origin/main

110d467f (dominance over n down, k up, d up, w down, with equal parameters counting as dominated), plus a floor kd^2/n >= 12; a candidate was dropped as soon as any side showed a logical lighter than the smallest record-making d for its (n, k, w).

Evidence trail

Every operator in the table was re-verified on the CPU with verify/gf2.py (zero syndrome against the opposite checks, outside the row space of its own checks, weight recounted) before it was recorded. Weights are the lightest found by that run.

| instrument | side | trials | seed | lightest logical | |---|---|---:|---:|---:| | screen stage 1 (estimate) | X | 100,000 | 71000081009 | 22 | | screen stage 2 (estimate) | X | 500,000 | 71000081300 | 22 | | screen stage 3 (recover) | X | 2,000,000 | 71000081600 | 22 | | screen stage 1 (estimate) | Z | 100,000 | 71000081010 | 22 | | screen stage 2 (estimate) | Z | 500,000 | 71000081301 | 22 | | screen stage 3 (recover) | Z | 2,000,000 | 71000081601 | 22 | | gf2_fast CPU RIS, pair depth 8, 1 thread, both sides | X | 200,000 | 876 | 22 | | finalist GPU deep kernel, full basis, pair depth 8 | X | 50,000,000 | 777 | 22 | | finalist GPU deep kernel, full basis, pair depth 8 | Z | 50,000,000 | 778 | 22 | | board fast pass, gf2_fast pair depth 8, 2 threads, both sides | both | 8,000,000 | 5360 | 22 |

Board CPU gate (verify/validate_candidate.py, then verify/gate_changed.py's structural and fast passes, run out of CI on the candidate file, laptop, 2 threads): validate_candidate passed = True (labels ['advances the weight-6 x unrestricted board', 'literature novelty UNVERIFIED']); circulant-GB structural pass lightest single-block logical 80 at 400,000 trials; gf2_fast fast pass lightest logical 22 at 8,000,000 trials (751 s, 2 threads, seed 5360); GATE passed.

Claim: witness-backed upper bound d <= 22 (X <= 22, Z <= 22), not exact. The lightest logical did not move between the 2,000,000-trial screen stage, the 200,000-trial CPU pass, the 50,000,000-trial deep-kernel pass per side, and the board's 8,000,000-trial fast pass.

Dead ends

• Cyclic GB with weight-3 polynomials: 12,288 draws over m in [101, 349]

gave 263 stage-3 survivors, but only 13 distinct (n, k) parameter sets, because weight-3 multiples of a divisor exist only for a few m (105, 127, 147, 155, 217, 231, 255, 279, 315, 341) and k is pinned at 2 deg g = 10 to 20; nothing reached kd^2/n 30.5, and the [[682,10,38]] and [[682,10,36]] screen survivors (kd^2/n 21.2, 19.0) fell to 34 under the board's gf2_fast pass and the finalist pass, where [[630,12,34]] dominates them.

• Weight-3 BB: 123,744 draws, k = 0 in 94 percent; the k >= 6 survivors are

low-rate codes at d 32 to 42 with kd^2/n 12 to 17.

• Metacyclic and dihedral 2BGA with |a| = |b| = 3: k = 0 for most draws

(71 percent of metacyclic draws); two metacyclic survivors in 0.4 GPU hours; the dihedral stream did not run (the pod's GPU was needed by another session).

• Ties: the same construction reproduces the board's [[254,14,16]] and

[[510,16,24]] parameter sets repeatedly; those are not records and were filtered out.

Tools

Generators from research/kit (search.py samplers, bb.py, group_algebra.py) and research/cyclic_gb.py; GPU RIS from verify/ris_gpu.cu built with nvcc for an NVIDIA A40 (blocking-sync host thread), driven through verify/ris_gpu.py's input format; gf2_fast from verify/ for exact k, kernels, and the CPU RIS; verify/validate_candidate.py and verify/gate_changed.py for the gate; a nauty canonical-form check of the typed Tanner graph (pynauty, outside this repo) for the novelty label. Model: Claude Fable 5.1 (Claude Code), matching provenance.model. Compute: about 2.4 GPU hours for the stream that produced this code, about 0.15 GPU hours for the finalist pass, and about 1 CPU hour for the gate.

Reproduction

Rebuild (H_X, H_Z) from the construction string in the code file's provenance: Cyclic (single-circulant-pair) generalized-bicycle code over Z_155 (research/cyclic_gb.py build_cyclic_gb): a(x) = 1 + x^4 + x^52, b(x) = 1 + x^19 + x^49; H_X = [circ(a) | circ(b)], H_Z = [circ(b)^T | circ(a)^T] with circ(v)[i, (i + j) mod 155] = v[j]. n = 2m = 310, k = 2 deg gcd(a, b, x^155 - 1) = 10, every check has weight 6. a and b were drawn as weight-3 multiples of a degree-5 divisor g of x^155 - 1 (g as a little-endian bit integer: 59).